This link http://mathworld.wolfram.com/GaltonBoard.html suggests that a certain specific placement of pegs can be used to simulate a binomial or a normal distribution. Is there a specific peg configuration that can be used to generate the uniform distribution. In general, can any standard distribution be simulated using configuration of pegs?
The method above to converge to the uniform distribution is in the right spirit to generalize the Galton board. A simple regular repeating pattern.
The following is much less aesthetic (and my crude illustration makes it worse) but does show how to get an exact uniform distribution, at least with $2^k$ cells.